A continuous Helson surface in \(R^ 3\) (Q793266)

For some time it has been known that there exist continuous Helson curves in \({\mathbb{R}}^ 2\). This result, which is related to Lusin's rearrangement problem, had been proved first by J. Kahane in 1968 with the aid of Baire category arguments. Later \textit{O. C. McGehee} and \textit{G. S. Woodward} [Ark. Mat. 20, 169-199 (1982; Zbl 0503.43005)] extended this result, giving a concrete construction of a Helson k-manifold in \({\mathbb{R}}^{nk}\) for \(n\geq k+1\). We present a construction of a Helson 2-manifold in \({\mathbb{R}}^ 3\). With modification, our method should even suffice to prove that there are Helson hypersurfaces in any \({\mathbb{R}}^ n\).